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Nucleon-Nucleon Interactions from the Quark Model C. Downum1∗, J. R. Stone∗,†, T. Barnes†,∗∗, E. S. Swanson‡ and I. Vidaña§ ∗ClarendonLaboratory,UniversityofOxford,ParksRoad,Oxford,OX13PU,UK †DepartmentofPhysicsandAstronomy,UniversityofTennessee,Knoxville,TN37996,USA ∗∗PhysicsDivision,OakRidgeNationalLaboratory,OakRidge,TN37831,USA 0 ‡DepartmentofPhysicsandAstronomy,UniversityofPittsburgh,Pittsburgh,PA15260,USA 1 §CentrodeFísicaComputacional.DepartamentodeFísica.UniversidadedeCoimbra,RuaLarga,3004-516 0 Coimbra,Portugal 2 n Abstract. We report on investigations of the applicability of non-relativistic constituent quark models to the low-energy a nucleon-nucleon (NN) interaction. The major innovations of a resulting NN potential are the use of the 3P decay model J 0 andquarkmodel wavefunctionstoderivenucleon-nucleon-meson form-factors,andtheuseofacoloredspin-spincontact 9 hyperfineinteractiontomodeltherepulsivecoreratherthanthephenomenologicaltreatmentcommoninotherNNpotentials. 1 WepresenttheresultsofthemodelforexperimentalfreeNNscatteringphaseshifts,S-wavescatteringlengthsandeffective rangesanddeuteronproperties.Plansforfuturestudyarediscussed. ] h Keywords: Nucleon-NucleonInteraction,QuarkModel,MesonExchange t PACS: 12.39.Jh,21.30.Fe,13.75.Cs - l c u INTRODUCTION n [ TheNNinteractionhasbeentheobjectofstudyforover50years.Thedevelopmentofthehighlyaccuratepotentialsin 1 1990sshowedthattheNNinteractioncanbequantitativelyexpressedverypreciselyalthoughthephysicalmechanism v of the interaction has yet to be understood. Most of these models use a partial-wave decomposition of the NN 0 potentialandvarypotentialformsandparameterizationsindependenceonpartialwavechannel.TheNN potentials 2 3 are either thoughtto rise fromone-bosonexchange(OBE) (e.g.Nijmegenand Bonn families) or are constructedof 3 differentfunctionalformsmultipliedbygeneraloperators(central,tensor,spin-orbitetc)whicharethenparameterized . (ArgonneandUrbanafamilies),withone-pionexchangeincludedtomodelthelong-rangepartofthepotential.Chiral 1 0 perturbationtheoryisamodernalternativeapproachwhichattempts,toderivetheNN-forcefromcontactinteractions 0 andpionexchangeasthelow-energyQCD degreesoffreedom.TheNijm I[1],Argonnev18[2],N3LO[3], andCD- 1 Bonn[4]potentialsareusedinthisworkforcomparison. : In thiswork,we presenta potentialmotivatedfromthe Non-RelativisticQuarkModel,the Oxfordpotential.It is v i showntobeabletowellreproduceNNscatteringphaseshiftsdataandthepropertiesofthedeuteron. X This work is organized as follows. First, we briefly review two quark models of strong interaction dynamics. r Secondly,wecreatetheOxfordpotentialusingthesemodels.Thentheresultsofourparametertuningarepresented a forNNdata andwe showthehighmomentumbehaviorof thepotential.Finally,we giveourconclusionsandplans forfuturework. QUARKMODELS OFSTRONG INTERACTIONS 3P Model 0 It was first suggested by Micu[5] that hadron decays might be dominated by the production of a qq¯ pair from the vacuum with JPC =0++ quantum numbers,i.e. they are in a 2S+1L = 3P state. Her idea has been applied to J 0 successfullyestimate3-pointverticeswithoneadditionalmesoninthefinalstate[6]. 1 PresentingAuthor = FIGURE1. Adiagramillustratingthetechniqueforcalculatingquarkmodelform-factorsforuseinmesonexchangepotentials. × × FIGURE2. ArepresentativediagramforthequarkmodelOGEpotential.The×denotethecoloredspin-spincontactinteractions. Fig. 1 illustrates how to parameterize the effective field theory from the microscopic decay diagram using the3P model.ExternalhadronsimpleharmonicoscillatorSHOwavefunctionsareattachedtothemicroscopicdecay 0 diagramstoaccountfornon-perturbativehadronizationeffectsandyieldsthefollowingexpressionsfortheform-factor F: 2 2 p~ +~p p~ −~p F(~p,p~ )=exp − f i − f i . (1) i f 24(3b 2+a 2) 6a 2 ( (cid:0) (cid:1) (cid:0) (cid:1) ) a ,b aretheSHOwavefunctionwidthsforthenucleonsandmesonrespectivelyandp, p aretheinitialandfinalstate i f momenta. OneGluonExchange Attempts to modelthe NN potential via a OGE interaction between boundstate hadronsdates back to Liberman in1977[7].ThepotentialduetotheOGEappearstobedominatedbythecontactspin-spinhyperfineinteraction(for recentreviewsseeShimizuetal.[8]andValcarceetal.[9]). Barneset al. used the “quarkBorndiagram”formalismto modelthe dominantcontactspin-spin interactionwith constituteinterchange(OGE+CI)to preservecolorneutrality.A typicaldiagramforthe processis presentin Fig. 2. TheOGE+CIinteractionisanalternativemodelofshortrangeNNdynamics–comparedwiththeusualheavymeson exchange model. Also, it illustrates that the different scales of physics can be self-consistently incorporated into a potentialusingthequarkmodel. THEOXFORDPOTENTIAL Forthe Oxfordpotential,we modeledthe NN interactionas a combinationofonepionexchange(OPE),onesigma exchange (OSE) and the OGE+CI potential. For the OPE and OSE, we use the standard the one meson exchange mechanism, with the form-factors calculated from the 3P model. Charge dependence was incorporated as in CD- 0 Bonn[4]. InordertofitthePandFwavephaseshifts,additionalrepulsionwasneededinthesechannels.Therefore,weadded oneomegaexchangeinthesechannelswithapurevector,isovectorcouplinginthesechannels.Adetailedpresentation ofthepotentialwillbethesubjectofaforthcomingpaper[10]andthequarkleveloriginsofthew exchangeeffects willbeinvestigated. 0 6 60 1S 100 3S -5 3D 5 0 1 1 40 75 -10 4 [deg] 20 50 -15 3 d -20 2 25 e 0 -25 1 1 0 -20 -30 0 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 T [MeV] T [MeV] T [MeV] T [MeV] lab lab lab lab 0 20 0 20 -10 1P1 10 3P0 -10 3P1 15 [deg]-20 0 -20 10 3P2 d -10 -30 -30 5 -20 -40 -30 -40 0 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 T [MeV] T [MeV] T [MeV] T [MeV] lab lab lab lab FIGURE3. PhaseshiftandinelasticitypredictionsoftheOxfordmodelfortheS,P,and3D partialwavesandthee mixing 1 1 angle.DatafromtheSAIDprogram[11]aretheblackdotswhilethepredictionsarethesolidredcurves. RESULTS The parameters of the Oxford potential were fitted to the single-energy values of the NN scattering phaseshifts as determinedfromtheSAIDanalysisprogram[11]andtothepropertiesofthedeuteron.Noattempthasbeenmadeto performa fit to NN scatteringdata ofthis ’proofof principle’potential.As the potentialdoesnotincludeCoulomb andhigher-orderelectromagneticcorrections,onlydatainthenpscatteringchannelhavebeenconsideredinthefit. Scattering PhaseShifts The NN potential is non-perturbatively strong. Therefore, we solve a partial wave decomposed Lippmann- Schwingertype equation.The phase shifts are extractedfrom the resulting T matrix using standardtechniques.We presenttheresultsofourphaseshiftparameterizationinFig. 3.Theresultsofthehighlyaccuratepotentialsarenot presented–allthepotentialsreproducetheexperimentalphaseshiftswell. AscanbeseenfromFig.3,theOxfordpotentialisabletoreproducethelow-lyingphaseshiftsverywell.Weonly considereddatauptoT =400MeV.HigherLphaseshiftsarenotpresentedhere,butachieveacomparablefit. Lab Deuteron Properties ThedeuteronistheonlyNNboundstate.WesolvetheSchrödingerEquationtocalculatetheboundstateproperties of the deuteron. The deuteron properties of the Oxford potential are presented along with the results of the other highlyaccuratepotentialsandtheexperimentalvalues.Detaileddiscussionoftheexperimentalvaluesmaybefound elsewhere[4]. The mass of the sigma was fine-tuned to reproduce the binding energy, the other properties are as followedwithoutsubsequentadjustment. Most quantitiesare accurate to within 1% when the experimentalerror is considered.Giventhe simplicity of the modelanditsconstraints,wefindtheseresultshighlyencouraging.Inparticularwedrawattentiontothequadrupole momentwhichtheOxfordpotentialisablebetterreproducethanthehighlyaccuratepotentials. TABLE1. ThedeuteronpropertiesoftheOxford,CD-Bonn,NijmegenI,andArgonnev18potentials alongwithexperimentalvalues.RelativisticandMesonCurrentcorrectionshavenotbeenincluded. Property Oxford CD-Bonn NijmegenI Argonnev18 N3LO Experiment Bd[MeV] 2.2246 2.224575 2.224575 2.224575 2.224575 2.224575±0.000009 AS[fm1/2] 0.8918 0.8846 0.8841 .8850 0.8843 0.8848±0.0009 h 0.0262 0.0256 0.0253 .0256 0.0256 0.0256±0.004 hr2i[fm] 1.9767 1.966 1.9666 1.967 1.978 1.971±0.006 pQd[fm2] 0.2871 0.270 0.2719 .270 0.275 0.28590±0.00030 PD[%] 5.604 4.85 5.664 5.76 4.51 S-waveScattering Lengths andEffective Ranges Our results for scattering lengths, aN, and effective ranges, rN, due to the strong nuclear force for the S-wave channels are summarized in Table 2 in comparison to experiment and predictions of the other highly accurate potentials. The 1S channels required the introduction of small charge symmetry breaking in addition to charge 0 independencebreaking.Inthe3S channel,thevaluesoftheaN andrN quotedwereobtainedusingparametersfitted 1 t t only to the binding energy of the deuteron. We find the discrepancy of the order of 1% between predictions of the Oxfordpotentialandexperimentencouraging. TABLE2. TheScatteringLengthsandEffectiveRangesfortheOxford,CD-Bonn,Nijmegen I[12],Argonnev18,andN3LOpotentialsalongwithexperimentalvalues. Oxford CD-Bonn NijmegenI Argonnev18 N3LO Experiment 1S 0 aN -17.365 -17.4602 -17.164 -17.083 pp rN 2.886 2.845 2.865 2.876 pp aN -18.949 -18.9680 -18.818 -18.900 -18.9±0.4 nn rN 2.859 2.819 2.834 2.838 2.75±0.11 nn aN -23.85 -23.7380 -23.084 -23.732 -23.74±0.020 np rN 2.753 2.671 2.703 2.752 2.77±0.05 np 3S 1 aN 5.46 5.4196 5.4194 5.402 5.417 5.419±0.007 t rN 1.77 1.751 1.7536 1.752 1.752 1.753±0.008 t HIGHMOMENTUM BEHAVIOR Ithasbeenknownforsometimethatthehighlyaccuratepotentialsremainstrongevenforhighvaluesofmomentum inducingstrongcorrelationsinmany-bodywavefunctions.Sophisticatedtechniqueshavebeendevelopedtointegrate outtheunphysicalhighmomentumdegreesoffreedomwhileretainingthephysical,lowmomentumbehaviorofthe potentials[13,14]. In Fig. 4 we plot the potential matrix elements in the S and triplet P channels as functions of momentum for the Oxford and Nijm I potentials. Because of the quark modelwave functionsthe Oxfordpotential has a non-local Gaussian form-factor which strongly suppresses the potential as momentum increases. This form-factor is similar to the one employed in the chiral N3LO potential. The rapid fall off in our potential, as opposed to say Nijm I, is physicallypreferred. CONCLUSIONS We have been conducting research into the possibility of calculating NN interactions mechanisms from the quark model.Todemonstratethatthistechniquecanleadtoanaccurate,physicallymotivatedpotential,wehaveproduced 30 30 20 20 1S 3P 0 0 3S 15 3P 15 15 1 15 2 10 10 3MeVfm] 0 0 13SS01 5 5 33PP02 V [ -15 -15 0 0 -5 -5 -30 -30 Oxford Nijm I Oxford Nijm I -10 -10 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 k [fm-1] k [fm-1] k [fm-1] k [fm-1] FIGURE4. TheOxfordandNijmIon-shellpotentialmatrixelementsforarangeofmomentum. theOxfordpotential. In this paper we reported on the preliminary predictions of the model. In addition to an excellent reproduction of the phase shifts, the potential was able to accurately reproducedeuteron and low-energyscattering propertiesto within 1%. Given the simplicity of this model,we find these results to be very encouraging,particularly as regards thequadrupolemomentofthedeuteron.Thepotentialisalsothefirsttoexhibitthecorrecthighmomentumbehavior resultingfromtheoreticallymotivatedform-factors. A paper which presents the Oxford potential in detail is under preparation by the authors[10]. Nuclear matter calculations have been carried out and show that the Oxford potential has a competitive equation of state. Future researchintothehighmomentumbehaviorofthepotentialisplanned. ACKNOWLEDGMENTS ResearchsponsoredinpartbytheOfficeofNuclearPhysics,U.S.DepartmentofEnergy.Wearegratefulforfruitful discussionswithStevenMoszkowski,RupertMachleidt,andArthurKerman. REFERENCES 1. V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. 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